1 (* Title: HOL/Library/Rational.thy
3 Author: Markus Wenzel, TU Muenchen
6 header {* Rational numbers *}
9 imports "../Presburger" GCD
13 subsection {* Rational numbers as quotient *}
15 subsubsection {* Construction of the type of rational numbers *}
18 ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
19 "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
21 lemma ratrel_iff [simp]:
22 "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
23 by (simp add: ratrel_def)
25 lemma refl_ratrel: "refl {x. snd x \<noteq> 0} ratrel"
26 by (auto simp add: refl_def ratrel_def)
28 lemma sym_ratrel: "sym ratrel"
29 by (simp add: ratrel_def sym_def)
31 lemma trans_ratrel: "trans ratrel"
32 proof (rule transI, unfold split_paired_all)
33 fix a b a' b' a'' b'' :: int
34 assume A: "((a, b), (a', b')) \<in> ratrel"
35 assume B: "((a', b'), (a'', b'')) \<in> ratrel"
36 have "b' * (a * b'') = b'' * (a * b')" by simp
37 also from A have "a * b' = a' * b" by auto
38 also have "b'' * (a' * b) = b * (a' * b'')" by simp
39 also from B have "a' * b'' = a'' * b'" by auto
40 also have "b * (a'' * b') = b' * (a'' * b)" by simp
41 finally have "b' * (a * b'') = b' * (a'' * b)" .
42 moreover from B have "b' \<noteq> 0" by auto
43 ultimately have "a * b'' = a'' * b" by simp
44 with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto
47 lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
48 by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
50 lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
51 lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
53 lemma equiv_ratrel_iff [iff]:
54 assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
55 shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"
56 by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)
58 typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"
60 have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp
61 then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)
64 lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat"
65 by (simp add: Rat_def quotientI)
67 declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]
70 subsubsection {* Representation and basic operations *}
73 Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
74 [code func del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
78 lemma Rat_cases [case_names Fract, cases type: rat]:
79 assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"
81 using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
83 lemma Rat_induct [case_names Fract, induct type: rat]:
84 assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"
86 using assms by (cases q) simp
89 shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b"
90 and "\<And>a. Fract a 0 = Fract 0 1"
91 and "\<And>a c. Fract 0 a = Fract 0 c"
92 by (simp_all add: Fract_def)
94 instantiation rat :: "{comm_ring_1, recpower}"
98 Zero_rat_def [code, code unfold]: "0 = Fract 0 1"
101 One_rat_def [code, code unfold]: "1 = Fract 1 1"
104 add_rat_def [code func del]:
105 "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
106 ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
108 lemma add_rat [simp]:
109 assumes "b \<noteq> 0" and "d \<noteq> 0"
110 shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
112 have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
114 by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
115 with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
119 minus_rat_def [code func del]:
120 "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
122 lemma minus_rat [simp, code]: "- Fract a b = Fract (- a) b"
124 have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
125 by (simp add: congruent_def)
126 then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
129 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
130 by (cases "b = 0") (simp_all add: eq_rat)
133 diff_rat_def [code func del]: "q - r = q + - (r::rat)"
135 lemma diff_rat [simp]:
136 assumes "b \<noteq> 0" and "d \<noteq> 0"
137 shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
138 using assms by (simp add: diff_rat_def)
141 mult_rat_def [code func del]:
142 "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
143 ratrel``{(fst x * fst y, snd x * snd y)})"
145 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
147 have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"
148 by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all
149 then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)
152 lemma mult_rat_cancel:
153 assumes "c \<noteq> 0"
154 shows "Fract (c * a) (c * b) = Fract a b"
156 from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
157 then show ?thesis by (simp add: mult_rat [symmetric])
162 rat_power_0: "q ^ 0 = (1\<Colon>rat)"
163 | rat_power_Suc: "q ^ Suc n = (q\<Colon>rat) * (q ^ n)"
166 fix q r s :: rat show "(q * r) * s = q * (r * s)"
167 by (cases q, cases r, cases s) (simp add: eq_rat)
169 fix q r :: rat show "q * r = r * q"
170 by (cases q, cases r) (simp add: eq_rat)
172 fix q :: rat show "1 * q = q"
173 by (cases q) (simp add: One_rat_def eq_rat)
175 fix q r s :: rat show "(q + r) + s = q + (r + s)"
176 by (cases q, cases r, cases s) (simp add: eq_rat ring_simps)
178 fix q r :: rat show "q + r = r + q"
179 by (cases q, cases r) (simp add: eq_rat)
181 fix q :: rat show "0 + q = q"
182 by (cases q) (simp add: Zero_rat_def eq_rat)
184 fix q :: rat show "- q + q = 0"
185 by (cases q) (simp add: Zero_rat_def eq_rat)
187 fix q r :: rat show "q - r = q + - r"
188 by (cases q, cases r) (simp add: eq_rat)
190 fix q r s :: rat show "(q + r) * s = q * s + r * s"
191 by (cases q, cases r, cases s) (simp add: eq_rat ring_simps)
193 show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
195 fix q :: rat show "q * 1 = q"
196 by (cases q) (simp add: One_rat_def eq_rat)
200 show "q ^ 0 = 1" by simp
201 show "q ^ (Suc n) = q * (q ^ n)" by simp
206 lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
207 by (induct k) (simp_all add: Zero_rat_def One_rat_def)
209 lemma of_int_rat: "of_int k = Fract k 1"
210 by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
212 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
213 by (rule of_nat_rat [symmetric])
215 lemma Fract_of_int_eq: "Fract k 1 = of_int k"
216 by (rule of_int_rat [symmetric])
218 instantiation rat :: number_ring
222 rat_number_of_def [code func del]: "number_of w = Fract w 1"
224 instance by intro_classes (simp add: rat_number_of_def of_int_rat)
228 lemma rat_number_collapse [code post]:
231 "Fract (number_of k) 1 = number_of k"
234 (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
236 lemma rat_number_expand [code unfold]:
239 "number_of k = Fract (number_of k) 1"
240 by (simp_all add: rat_number_collapse)
242 lemma iszero_rat [simp]:
243 "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
244 by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
246 lemma Rat_cases_nonzero [case_names Fract 0]:
247 assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
248 assumes 0: "q = 0 \<Longrightarrow> C"
250 proof (cases "q = 0")
251 case True then show C using 0 by auto
254 then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
255 moreover with False have "0 \<noteq> Fract a b" by simp
256 with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
257 with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
262 subsubsection {* The field of rational numbers *}
264 instantiation rat :: "{field, division_by_zero}"
268 inverse_rat_def [code func del]:
269 "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
270 ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
272 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
274 have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
275 by (auto simp add: congruent_def mult_commute)
276 then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
280 divide_rat_def [code func del]: "q / r = q * inverse (r::rat)"
282 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
283 by (simp add: divide_rat_def)
286 show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
287 (simp add: rat_number_collapse)
290 assume "q \<noteq> 0"
291 then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
292 (simp_all add: mult_rat inverse_rat rat_number_expand eq_rat)
295 show "q / r = q * inverse r" by (simp add: divide_rat_def)
301 subsubsection {* Various *}
303 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
304 by (simp add: rat_number_expand)
306 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
307 by (simp add: Fract_of_int_eq [symmetric])
309 lemma Fract_number_of_quotient [code post]:
310 "Fract (number_of k) (number_of l) = number_of k / number_of l"
311 unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
313 lemma Fract_1_number_of [code post]:
314 "Fract 1 (number_of k) = 1 / number_of k"
315 unfolding Fract_of_int_quotient number_of_eq by simp
317 subsubsection {* The ordered field of rational numbers *}
319 instantiation rat :: linorder
323 le_rat_def [code func del]:
324 "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
325 {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
328 assumes "b \<noteq> 0" and "d \<noteq> 0"
329 shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
331 have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
333 proof (clarsimp simp add: congruent2_def)
334 fix a b a' b' c d c' d'::int
335 assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
336 assume eq1: "a * b' = a' * b"
337 assume eq2: "c * d' = c' * d"
339 let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
341 fix a b c d x :: int assume x: "x \<noteq> 0"
342 have "?le a b c d = ?le (a * x) (b * x) c d"
344 from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
346 ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
347 by (simp add: mult_le_cancel_right)
348 also have "... = ?le (a * x) (b * x) c d"
349 by (simp add: mult_ac)
350 finally show ?thesis .
352 } note le_factor = this
354 let ?D = "b * d" and ?D' = "b' * d'"
355 from neq have D: "?D \<noteq> 0" by simp
356 from neq have "?D' \<noteq> 0" by simp
357 hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
359 also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
360 by (simp add: mult_ac)
361 also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
362 by (simp only: eq1 eq2)
363 also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
364 by (simp add: mult_ac)
365 also from D have "... = ?le a' b' c' d'"
366 by (rule le_factor [symmetric])
367 finally show "?le a b c d = ?le a' b' c' d'" .
369 with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
373 less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
375 lemma less_rat [simp]:
376 assumes "b \<noteq> 0" and "d \<noteq> 0"
377 shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
378 using assms by (simp add: less_rat_def eq_rat order_less_le)
383 assume "q \<le> r" and "r \<le> s"
385 proof (insert prems, induct q, induct r, induct s)
386 fix a b c d e f :: int
387 assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
388 assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
389 show "Fract a b \<le> Fract e f"
391 from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
392 by (auto simp add: zero_less_mult_iff linorder_neq_iff)
393 have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
395 from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
397 with ff show ?thesis by (simp add: mult_le_cancel_right)
399 also have "... = (c * f) * (d * f) * (b * b)" by algebra
400 also have "... \<le> (e * d) * (d * f) * (b * b)"
402 from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
404 with bb show ?thesis by (simp add: mult_le_cancel_right)
406 finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
407 by (simp only: mult_ac)
408 with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
409 by (simp add: mult_le_cancel_right)
410 with neq show ?thesis by simp
414 assume "q \<le> r" and "r \<le> q"
416 proof (insert prems, induct q, induct r)
418 assume neq: "b \<noteq> 0" "d \<noteq> 0"
419 assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
420 show "Fract a b = Fract c d"
422 from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
424 also have "... \<le> (a * d) * (b * d)"
426 from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
428 thus ?thesis by (simp only: mult_ac)
430 finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
431 moreover from neq have "b * d \<noteq> 0" by simp
432 ultimately have "a * d = c * b" by simp
433 with neq show ?thesis by (simp add: eq_rat)
439 show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
440 by (induct q, induct r) (auto simp add: le_less mult_commute)
441 show "q \<le> r \<or> r \<le> q"
442 by (induct q, induct r)
443 (simp add: mult_commute, rule linorder_linear)
449 instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
453 abs_rat_def [code func del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
455 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
456 by (auto simp add: abs_rat_def zabs_def Zero_rat_def less_rat not_less le_less minus_rat eq_rat zero_compare_simps)
459 sgn_rat_def [code func del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
461 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
462 unfolding Fract_of_int_eq
463 by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
464 (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
467 "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
470 "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
472 instance by intro_classes
473 (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
477 instance rat :: ordered_field
480 show "q \<le> r ==> s + q \<le> s + r"
481 proof (induct q, induct r, induct s)
482 fix a b c d e f :: int
483 assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
484 assume le: "Fract a b \<le> Fract c d"
485 show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
487 let ?F = "f * f" from neq have F: "0 < ?F"
488 by (auto simp add: zero_less_mult_iff)
489 from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
491 with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
492 by (simp add: mult_le_cancel_right)
493 with neq show ?thesis by (simp add: mult_ac int_distrib)
496 show "q < r ==> 0 < s ==> s * q < s * r"
497 proof (induct q, induct r, induct s)
498 fix a b c d e f :: int
499 assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
500 assume le: "Fract a b < Fract c d"
501 assume gt: "0 < Fract e f"
502 show "Fract e f * Fract a b < Fract e f * Fract c d"
504 let ?E = "e * f" and ?F = "f * f"
505 from neq gt have "0 < ?E"
506 by (auto simp add: Zero_rat_def order_less_le eq_rat)
507 moreover from neq have "0 < ?F"
508 by (auto simp add: zero_less_mult_iff)
509 moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
511 ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
512 by (simp add: mult_less_cancel_right)
513 with neq show ?thesis
514 by (simp add: mult_ac)
519 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
520 assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
523 have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
525 fix a::int and b::int
527 hence "0 < -b" by simp
528 hence "P (Fract (-a) (-b))" by (rule step)
529 thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
532 thus "P q" by (force simp add: linorder_neq_iff step step')
535 lemma zero_less_Fract_iff:
536 "0 < b ==> (0 < Fract a b) = (0 < a)"
537 by (simp add: Zero_rat_def order_less_imp_not_eq2 zero_less_mult_iff)
540 subsection {* Arithmetic setup *}
543 declaration {* K rat_arith_setup *}
546 subsection {* Embedding from Rationals to other Fields *}
548 class field_char_0 = field + ring_char_0
550 subclass (in ordered_field) field_char_0 ..
555 definition of_rat :: "rat \<Rightarrow> 'a" where
556 [code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
560 lemma of_rat_congruent:
561 "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
562 apply (rule congruent.intro)
563 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
564 apply (simp only: of_int_mult [symmetric])
567 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
568 unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
570 lemma of_rat_0 [simp]: "of_rat 0 = 0"
571 by (simp add: Zero_rat_def of_rat_rat)
573 lemma of_rat_1 [simp]: "of_rat 1 = 1"
574 by (simp add: One_rat_def of_rat_rat)
576 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
577 by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
579 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
580 by (induct a, simp add: of_rat_rat)
582 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
583 by (simp only: diff_minus of_rat_add of_rat_minus)
585 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
586 apply (induct a, induct b, simp add: of_rat_rat)
587 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
590 lemma nonzero_of_rat_inverse:
591 "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
592 apply (rule inverse_unique [symmetric])
593 apply (simp add: of_rat_mult [symmetric])
596 lemma of_rat_inverse:
597 "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
599 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
601 lemma nonzero_of_rat_divide:
602 "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
603 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
606 "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
607 = of_rat a / of_rat b"
608 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
611 "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
612 by (induct n) (simp_all add: of_rat_mult power_Suc)
614 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
615 apply (induct a, induct b)
616 apply (simp add: of_rat_rat eq_rat)
617 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
618 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
622 "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
623 proof (induct r, induct s)
625 assume not_zero: "b > 0" "d > 0"
626 then have "b * d > 0" by (rule mult_pos_pos)
627 have of_int_divide_less_eq:
628 "(of_int a :: 'a) / of_int b < of_int c / of_int d
629 \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
630 using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
631 show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
632 \<longleftrightarrow> Fract a b < Fract c d"
633 using not_zero `b * d > 0`
634 by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
635 (auto intro: mult_strict_right_mono mult_right_less_imp_less)
638 lemma of_rat_less_eq:
639 "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
640 unfolding le_less by (auto simp add: of_rat_less)
642 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
644 lemma of_rat_eq_id [simp]: "of_rat = id"
647 show "of_rat a = id a"
649 (simp add: of_rat_rat Fract_of_int_eq [symmetric])
652 text{*Collapse nested embeddings*}
653 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
654 by (induct n) (simp_all add: of_rat_add)
656 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
657 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
659 lemma of_rat_number_of_eq [simp]:
660 "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
661 by (simp add: number_of_eq)
663 lemmas zero_rat = Zero_rat_def
664 lemmas one_rat = One_rat_def
667 rat_of_nat :: "nat \<Rightarrow> rat"
669 "rat_of_nat \<equiv> of_nat"
672 rat_of_int :: "int \<Rightarrow> rat"
674 "rat_of_int \<equiv> of_int"
676 subsection {* The Set of Rational Numbers *}
682 Rats :: "'a set" where
683 [code func del]: "Rats = range of_rat"
690 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
691 by (simp add: Rats_def)
693 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
694 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
696 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
697 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
699 lemma Rats_number_of [simp]:
700 "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
701 by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
703 lemma Rats_0 [simp]: "0 \<in> Rats"
704 apply (unfold Rats_def)
705 apply (rule range_eqI)
706 apply (rule of_rat_0 [symmetric])
709 lemma Rats_1 [simp]: "1 \<in> Rats"
710 apply (unfold Rats_def)
711 apply (rule range_eqI)
712 apply (rule of_rat_1 [symmetric])
715 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
716 apply (auto simp add: Rats_def)
717 apply (rule range_eqI)
718 apply (rule of_rat_add [symmetric])
721 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
722 apply (auto simp add: Rats_def)
723 apply (rule range_eqI)
724 apply (rule of_rat_minus [symmetric])
727 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
728 apply (auto simp add: Rats_def)
729 apply (rule range_eqI)
730 apply (rule of_rat_diff [symmetric])
733 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
734 apply (auto simp add: Rats_def)
735 apply (rule range_eqI)
736 apply (rule of_rat_mult [symmetric])
739 lemma nonzero_Rats_inverse:
740 fixes a :: "'a::field_char_0"
741 shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
742 apply (auto simp add: Rats_def)
743 apply (rule range_eqI)
744 apply (erule nonzero_of_rat_inverse [symmetric])
747 lemma Rats_inverse [simp]:
748 fixes a :: "'a::{field_char_0,division_by_zero}"
749 shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
750 apply (auto simp add: Rats_def)
751 apply (rule range_eqI)
752 apply (rule of_rat_inverse [symmetric])
755 lemma nonzero_Rats_divide:
756 fixes a b :: "'a::field_char_0"
757 shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
758 apply (auto simp add: Rats_def)
759 apply (rule range_eqI)
760 apply (erule nonzero_of_rat_divide [symmetric])
763 lemma Rats_divide [simp]:
764 fixes a b :: "'a::{field_char_0,division_by_zero}"
765 shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
766 apply (auto simp add: Rats_def)
767 apply (rule range_eqI)
768 apply (rule of_rat_divide [symmetric])
771 lemma Rats_power [simp]:
772 fixes a :: "'a::{field_char_0,recpower}"
773 shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
774 apply (auto simp add: Rats_def)
775 apply (rule range_eqI)
776 apply (rule of_rat_power [symmetric])
779 lemma Rats_cases [cases set: Rats]:
780 assumes "q \<in> \<rat>"
781 obtains (of_rat) r where "q = of_rat r"
784 from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
785 then obtain r where "q = of_rat r" ..
789 lemma Rats_induct [case_names of_rat, induct set: Rats]:
790 "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
791 by (rule Rats_cases) auto
794 subsection {* Implementation of rational numbers as pairs of integers *}
796 lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
797 proof (cases "a = 0 \<or> b = 0")
798 case True then show ?thesis by (auto simp add: eq_rat)
801 case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
802 then have "?c \<noteq> 0" by simp
803 then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
804 moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
805 by (simp add: times_div_mod_plus_zero_one.mod_div_equality)
806 moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
807 moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
808 ultimately show ?thesis
809 by (simp add: mult_rat [symmetric])
812 definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
813 [simp, code func del]: "Fract_norm a b = Fract a b"
815 lemma [code func]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
816 if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
817 by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
820 "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
821 by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
823 instantiation rat :: eq
826 definition [code func del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
828 instance by default (simp add: eq_rat_def)
830 lemma rat_eq_code [code]:
831 "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
832 then c = 0 \<or> d = 0
834 then a = 0 \<or> b = 0
836 by (auto simp add: eq eq_rat)
841 assumes "b \<noteq> 0"
843 shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
845 have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
846 have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
847 proof (cases "b * d > 0")
849 moreover from True have "sgn b * sgn d = 1"
850 by (simp add: sgn_times [symmetric] sgn_1_pos)
851 ultimately show ?thesis by (simp add: mult_le_cancel_right)
853 case False with assms have "b * d < 0" by (simp add: less_le)
854 moreover from this have "sgn b * sgn d = - 1"
855 by (simp only: sgn_times [symmetric] sgn_1_neg)
856 ultimately show ?thesis by (simp add: mult_le_cancel_right)
858 also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
859 by (simp add: abs_sgn mult_ac)
860 finally show ?thesis using assms by simp
864 assumes "b \<noteq> 0"
866 shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
868 have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
869 have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
870 proof (cases "b * d > 0")
872 moreover from True have "sgn b * sgn d = 1"
873 by (simp add: sgn_times [symmetric] sgn_1_pos)
874 ultimately show ?thesis by (simp add: mult_less_cancel_right)
876 case False with assms have "b * d < 0" by (simp add: less_le)
877 moreover from this have "sgn b * sgn d = - 1"
878 by (simp only: sgn_times [symmetric] sgn_1_neg)
879 ultimately show ?thesis by (simp add: mult_less_cancel_right)
881 also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
882 by (simp add: abs_sgn mult_ac)
883 finally show ?thesis using assms by simp
886 lemma rat_less_eq_code [code]:
887 "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
888 then sgn c * sgn d \<ge> 0
890 then sgn a * sgn b \<le> 0
891 else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
892 by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
893 (auto simp add: sgn_times sgn_0_0 le_less sgn_1_pos [symmetric] sgn_1_neg [symmetric])
895 lemma rat_le_eq_code [code]:
896 "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
897 then sgn c * sgn d > 0
899 then sgn a * sgn b < 0
900 else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
901 by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
902 (auto simp add: sgn_times sgn_0_0 sgn_1_pos [symmetric] sgn_1_neg [symmetric],
903 auto simp add: sgn_1_pos)
905 lemma rat_plus_code [code]:
906 "Fract a b + Fract c d = (if b = 0
910 else Fract_norm (a * d + c * b) (b * d))"
911 by (simp add: eq_rat, simp add: Zero_rat_def)
913 lemma rat_times_code [code]:
914 "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
917 lemma rat_minus_code [code]:
918 "Fract a b - Fract c d = (if b = 0
922 else Fract_norm (a * d - c * b) (b * d))"
923 by (simp add: eq_rat, simp add: Zero_rat_def)
925 lemma rat_inverse_code [code]:
926 "inverse (Fract a b) = (if b = 0 then Fract 1 0
927 else if a < 0 then Fract (- b) (- a)
929 by (simp add: eq_rat)
931 lemma rat_divide_code [code]:
932 "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
935 hide (open) const Fract_norm
937 text {* Setup for SML code generator *}
942 fun term_of_rat (p, q) =
944 val rT = Type ("Rational.rat", [])
946 if q = 1 orelse p = 0 then HOLogic.mk_number rT p
947 else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
948 HOLogic.mk_number rT p $ HOLogic.mk_number rT q
954 val p = random_range 0 i;
955 val q = random_range 1 (i + 1);
956 val g = Integer.gcd p q;
959 val r = (if one_of [true, false] then p' else ~ p',
960 if p' = 0 then 0 else q')
962 (r, fn () => term_of_rat r)
970 "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
972 fun rat_of_int 0 = (0, 0)
973 | rat_of_int i = (i, 1);